Some of the most simple and obvious advantages of Breastfeeding are:
It's free, always the right temperature, requires no sterilisation, produces less smelly nappies, less constipation and can also help baby fight off infections - this last one is especially useful if the baby is in a nursery or similar.

However, there are many thoroughly researched medical benefits of breastfeeding too - here are some of them.

Breastmilk contains antibodies to protect your baby from infection and allergy:
Sickness and Diarrhoea (gastroenteritis, which may be v severe), chest infections, ear infections and urine infections are all more likely in bottle fed babies. Bottle fed babies are FIVE TIMES more likely to develop gastro-intestinal illness (Howie et al, 1990) than BF babies. Interestingly, breastfeeding exclusively for 13 or more weeks, protects BEYOND the period of breastfeeding itself - and the longer you feed, the greater the period of extension, it seems. Respiratory Infections. According to Wilson et al (1998), bottle fed infants are at almost twice the risk of developing respiratory illness at any time during the first 7 years of life (versus BF to OVER 15 weeks). The study also found that solid feeding before 15 weeks was associated with increased likelihood of wheeze during childhood, as well as increased percentage bodyfat and weight in childhood. Allergic Disease: Saarisen & Kajosaari (1995) found that infants BF for longer than one month without other milk supplements had significantly less food allergy at three years old, and significantly less respiratory allergy at SEVENTEEN YEARS old! Those that were BF for more than 6 months had significantly less eczema during the first three years, and less allergies in adolescence. (all info taken from the UNICEF Baby Friendly Initiative)

Babies Breastfed exclusively for 3 mths are less likely to get Insulin Dependent Diabetes:
(Gerstein (1994)) Insulin Dependent Diabetes patients were more likely to have been exposed to cow's milk protein before 4 months. The researchers estimated that 30% of type one diabetes could be prevented by removing cow's milk products from the diet of 90% of the population in the first three months of life. Early onset IDD patients were more likely than healthy controls to have been breastfed for less than 3 months - or not at all.

Children who are breastfed are less likely to be near-sighted:
Researchers in Singapore found that an acid, known as DHA, which is found in breast milk, could improve early visual development in babies. According to the American Academy of Family Physicians, nearsightedness affects as much as 40 percent of the population. Scientists also found that DHA is important in the development of cells in the retina.

Breastfeeding decreases the risk of Ovarian Cancer:
Rosenblatt et al (1993) found that there was a 20-25% decrease in the risk of ovarian cancer among women who breastfed for at least 2 months per pregnancy.Some factors decrease ovarian cancer risk:
Any factor that inhibits ovulation (release of an egg from the ovary) seems to protect against development of ovarian cancer. This may be because ovulation disrupts the epithelial layer of the ovary. As cells divide to repair the damage, uncontrolled division and malignant changes may occur.
Term pregnancy (lasting the full 9 months) significantly reduces the risk of ovarian cancer. As the number of pregnancies increases, the risk of ovarian cancer decreases.
Use of oral contraceptives (birth control pills) reduces the risk of ovarian cancer.
Breastfeeding lowers risk of ovarian cancer, and the risk decreases with increasing duration of breastfeeding.

Breastfeeding is linked to increased Neurological Development in the baby:
(Anderson et al 1999) Meta-analysis from 20 studies found that BF was associated with significantly higher scores for cognitive development and that the development of benefits increased with duration of feeding. Lucas et al (1992) Children who had been born pre-term were studied at age 7-8 years. Those who had been fed breastmilk in the early weeks of life had an average 8.3 point advantage in IQ over those who had received artificial baby milk. Morrow-Tlucak et al (1988) Measured cognitive development in children at age 2 years. Those BF for 4 months had an average 3.7 point advantage over those artificially fed. Those fed for over 4 months had an average 9.1 point advantage. Vestergaard et al (1999) Examined infants before 1 year of age to reduce the role of enviromental influence. The proportion of infants who mastered specific milestones increased consistently with increasing duration of BF.

Hip Fractures and Bone Density:
Cumming and Klineberg (1993) found that parous women (mothers) who had not breastfed had twice the risk of hip fracture over the age of 65 years as nulliparous (barren) women, and those who had breastfed. Polette et al (1999) found that among women who had BF FULLY for 6 months, bone mineral density decreased during this time but had increased by 18 months post partum to a higher level than before.

More research on breastfeeding lowering the risk of pre-menopausal breast cancer:
The UKNCC Study Group (1993) found that the risk of developing breast cancer before the age of 36 was reduced both by the duration of breastfeeding, and the number of babies breastfed. Newcomb et al (1994) Breast cancer risk found to be 22% lower among pre-menopausal women who had ever breastfed than among those who had not. This study estimated that if all women breastfed for 4-12 months, breast cancer among pre-menopausal women could be reduced by as much as 11%, and if women with children breastfed for a lifetime total of 24 months or longer, the incidence of this form of cancer may be reduced by almost 25%.Breastfeeding for two or more years reduces a woman's risk of developing breast cancer by 50 percent, according to a study conducted by a Yale researcher among women in China. What he found was a 50 percent reduction in breast cancer risk among those women who breastfed for more than 24 months per child, compared to women who breastfed their children for less than 12 months. Zheng said the breastfeeding for more than 24 months also has a long term effect. It was found not only to reduce the risk of breast cancer diagnosed among pre-menopausal women, it was also found to reduce the riskof breast cancer among post-menopausal women as well. However, the age of first breastfeeding and the total number of children breastfed did not seem to have a significant impact on the breast cancer risk.

BEING breastfed can also reduce the risk of breast cancer: Children who are artificially fed or breastfed for only 6 months or less, are at an increased risk of developing cancer before age 15. The risk of artificially fed children was 1-8 times that of long-term breastfed children, and the risk for short-term feeders was 1-9 times that of long- term breast feeders.
Source: Davis, M.K. Infant Feeding and Childhood Cancer. "Lancet 1988

Others studies have shown that for both premenopausal and postmenopausal breast cancer, women who were breastfed as children, even if only for a short time, had a 25% lower risk of developing breast cancer than women who were bottle-fed as infants
Sources: Schwartzbaum, J. et al. An exploratory study of environmental and medical factors potentially related to childhood cancer. Med & Pediat Oncology 1991; 19(2): 115-21. Freudenheim, J. et al. 1994 Exposure to breast milk in infancy and the risk of breast cancer. Epidemiology 5:324-331.

Breastfeeding provides all a baby needs to develop and thrive for the first six months of life. By around six months, a baby can have other foods and drinks as well as breastmilk.
Breastmilk is always available fresh, in the right amounts and at the right temperature.
Breastmilk is a living fluid containing active cells that mop up bacteria and viruses as well as antibodies tailored to fight the infections the baby comes into contact with. These factors are entirely missing from artificial substitutes.
Breastfeeding provides a baby with a boost to their immune system just when they need it as their own is still developing.

Breastfeeding is much cheaper than bottle feeding for families.
Breastfeeding is good for the environment, it does not need processing, packaging, transporting or manufacturing.
More evidence about the differences in health outcomes for formula fed babies is published every year. Although it is not possible to carry out strictly controlled studies it is clear that breastfeeding makes a big difference to babies' health on average.

The use of formula instead of breastfeeding in industrialised countries is associated with:

A five times higher risk of admission to hospital with diarrhoea and vomiting as a baby.
More and more severe respiratory infections.
Greater risk of ear and urinary tract infections.
Lower average scores on tests of neurological development.
Increased risk of allergies and greater intensity of problems from allergies.
Increased risk of develpoing insulin-dependent diabetes as a child.
Greater risk of overweight and higher blood pressure as children.
Increased risk of breast cancer in mothers who don't breastfeed.

This does not mean that all, or even most, formula-fed babies are sick and all breastfed babies are healthy. This information comes from large scale studies in a number of different countries.